The Navier--Stokes equations are axially symmetric, so with symmetric boundary conditions, how can features such as Karman vortex streets develop?

I understand that in reality symmetry does never completely hold due to minimal fluctuations of geometries, densities, etc., but in the linked article, vortex streets develop even in numerical simulations

  • The von Karman vortex streets will not develop if you have a symmetry boundary condition at the center of your cylinder, but only if you mesh your entire cylinder. Furthermore, the Navier-Stokes equations are invariant by rotation (because of the symmetry of the stress tensor), but remember that they are a set of non-linear equations (velocity advects velocity), this can lead to all sorts of non-linear phenomena such as the vortex shedding encountered in the vortex streets. – BlaB Aug 9 at 13:27
  • @BlaB so the assymmetry in this video youtube.com/watch?v=vBZDR0gMEEI develops because either the meshing or the boundary conditions were done asymmetrically? – Bananach Aug 9 at 13:59
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    No. in this case you have no symmetry boundary condition on the center of the cylinder (well circle). The circle is fully meshed. The instability develops because of the non-linearity of the Navier-Stokes equation. For a given Reynolds number, the solution without oscillations around the cylinder is unstable from a stability analysis point of view. Consequently, slight perturbations in the numerical solution (round-off errors, meshing, order of operations) start-off the oscillations. – BlaB Aug 9 at 14:11
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    A good example of stability analysis of a flow is the Rayleigh-Bénard instability. You can demonstrate via linear stability analysis that for a given critical Rayleigh number the absence of flow is unstable to any slight perturbation $\epsilon$. As such any type of rounding error in your numerical scheme is enough to generate this $\epsilon$ and start-off your Rayleigh-Bénard instability. – BlaB Aug 9 at 14:12
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    I am not sure what you mean by a symmetric numerical method. Somewhere somehow you will always do operations in a certain order, whether it is when solving a numerical system by an iterative (or direct) solver or by looping over certain cell in a certain order. This leads to some roundoff errors which are on the order of machine precision (say 1e-16). You could have a symmetric boundary condition at the center of the cylinder by meshing half the cylinder and imposing slip on that surface. In this way you would constraint your space to half a cylinder and the instability would not appear. – BlaB Aug 9 at 14:20

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